First of all, there is no starting point for the soul; here's what Krishna tells Arjuna in the Bhagavad Gita:
Never was there a time when I did not exist, nor you, nor all these kings; nor in the future shall any of us cease to be.... For the soul there is neither birth nor death at any time. He has not come into being, does not come into being, and will not come into being. He is unborn, eternal, ever-existing and primeval.
Now as to your claim that we only have seven births to get Moksha, that's not true, it can take as many births as necessary. That's why Krishna tells Arjuna in the Bhagavad Gita "Many, many births both you and I have passed." And similarly he says this later on in the Bhagavad Gita:
And when the yogī engages himself with sincere endeavor in making further progress, being washed of all contaminations, then ultimately, achieving perfection after many, many births of practice, he attains the supreme goal.
In any case, other than the seven chances part your description is largely accurate. As I discuss in this answer, the Garuda Paruna describes in great detail what happens in the afterlife:
These followers of [Yama] the King of Justice know accurately all the virtues and vices of mankind, and the karma born of mind, speech and body. To the man who pleases them by austerity, charity and truthful speech, they become benevolent, granting heaven and liberation. Knowing the wicked actions of the sinful, those truth-speakers, relating them before the King of Justice, become dispensers of misery.... Then the cruel messengers, having beaten them, say, "Go along, you sinner, to the very dreadful terrifying hells."
So as you can see, some people who do good deeds go to Devaloka, other people who do good deeds get Moksha, and people who do bad deeds go to Asuraloka.
And of course going to Devaloka and Asuraloka isn't where the process ends; concerning good people the Garuda Purana says "The righteous man having enjoyed heaven, is born in a stainless family." And concerning bad people it says "Those who are very sinful, having passed through dreadful hells produced by their great sins, are born here upon the exhaustion of their karma."
EDIT: You also asked "Yes, a soul exchanges bodies, but when does a soul take birth and how will fate be written to it (as it has no karma [no previous life actions recorded])." Adi Shankaracharya, in his commentary on the Brahma Sutras, says this isn't problematic because the actions of the soul go infinitely far back:
[F]or although the activity of the soul is not independent, yet the soul does act. The Lord indeed causes it to act, but it acts itself. Moreover, the Lord in causing it to act now has regard to its former efforts, and he caused it to act in a former existence, having regard to its efforts previous to that existence; a regressus against which, considering the eternity of the samsâra, no objections can be raised.